Solution of nonlinear mixed integral equation via collocation method basing on orthogonal polynomials
1Department of Mathematics, Faculty of Applied Science, Umm Al- Qura University, Makkah, Kingdom of Saudi Arabia.
Heliyon
|December 6, 2022
Summary
This study proves a unique solution for nonlinear mixed integral equations (NMIEs) and analyzes its stability. Numerical methods using Hermite and Laguerre polynomials provide accurate results and error estimates for various NMIE applications.
Area of Science:
- Applied Mathematics
- Numerical Analysis
- Integral Equations
Background:
- Nonlinear mixed integral equations (NMIEs) present complex challenges in mathematical physics and engineering.
- Understanding the existence and stability of NMIE solutions is crucial for accurate modeling.
Purpose of the Study:
- To establish the existence and uniqueness of a solution for a third-kind NMIE in position and time.
- To investigate the stability of the obtained solution.
- To develop and analyze numerical methods for solving NMIEs.
Main Methods:
- Transformation of the NMIE into a system of nonlinear integral equations (NIEs) using a quadratic method.
- Application of a collocation method with Hermite and Laguerre polynomials to derive nonlinear algebraic systems (NAS).
- Error estimation for the developed numerical techniques.
Main Results:
- Demonstrated the existence and uniqueness of a solution for the studied NMIE.
- Developed two distinct NAS based on Hermite and Laguerre polynomials.
- Computed numerical results and calculated error estimates for various NMIE types.
Conclusions:
- The proposed numerical methods, utilizing Hermite and Laguerre polynomials, effectively solve NMIEs.
- The study provides a robust framework for analyzing the existence, uniqueness, and stability of NMIE solutions.
Keywords:
Hermite polynomialsLaguerre polynomialsNonlinear algebraic systemNonlinear mixed integral equationMore Related Videos
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